Abstract Excision and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>ℓ</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:math> -Homology

We use the abstract setting of excisive functors in the language of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>∞</mml:mi> </mml:math> -categories to show that the best approximation to the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>ℓ</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:math> -homology functor by an excisive functor is trivial. Then we make an effort to explain the used language on a conceptual level for those who do not feel at home with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>∞</mml:mi> </mml:math> -categories, prove that the singular chain complex functor is indeed excisive in the abstract sense, and show how the latter leads to classical excision statements in the form of Mayer–Vietoris sequences.